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Uniqueness of locally stable Gibbs point processes via spatial birth-death dynamics
Probability · Submitted 1 Jul 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that for every locally stable and tempered pair potential $\phi$ with bounded range, there exists a unique infinite-volume Gibbs point process on $\mathbb{R}^d$ for every activity $\lambda < (e^{L} \hat{C}_{\phi})^{-1}$, where $L$ is the local stability constant and $\hat{C}_{\phi}:= \mathrm{sup}_{x \in \mathbb{R}^{d}} \int_{\mathbb{R}^{d}} 1 - e^{-|\phi(x, y)|} dy$ is the (weak) temperedness constant.
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We prove that for every locally stable and tempered pair potential $\phi$ with bounded range, there exists a unique infinite-volume Gibbs point process on $\mathbb{R}^d$ for every activity $\lambda < (e^{L} \hat{C}_{\phi})^{-1}$, where $L$ is the local stability constant and $\hat{C}_{\phi}:= \mathrm{sup}_{x \in \mathbb{R}^{d}} \int_{\mathbb{R}^{d}} 1 - e^{-|\phi(x, y)|} dy$ is the (weak) temperedness constant. Our result extends the uniqueness regime that is given by the classical Ruelle--Penrose bound by a factor of at least $e$, where the improvements becomes larger as the negative parts of the potential become more prominent (i.e., for attractive interactions at low temperature). Our technique is based on the approach of Dyer et al. (Rand. Struct. & Alg. '04): we show that for any bounded region and any boundary condition, we can construct a Markov process (in our case spatial birth-death dynamics) that converges rapidly to the finite-volume Gibbs point process while effects of the boundary condition propagate sufficiently slowly. As a result, we obtain a spatial mixing property that implies uniqueness of the infinite-volume Gibbs measure.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove that for every locally stable and tempered pair potential $\phi$ with bounded range, there exists a unique infinite-volume Gibbs point process on $\mathbb{R}^d$ for every activity $\lambda < (e^{L} \hat{C}_{\phi})^{-1}$, where $L$ is the local stability constant and $\hat{C}_{\phi}:= \mathrm{sup}_{x \in \mathbb{R}^{d}} \int_{\mathbb{R}^{d}} 1 - e^{-|\phi(x, y)|} dy$ is the (weak) temperedness constant. | result 0.97 | 0.96 | 0.93 | asserted 1.00 |
| s1 | Our result extends the uniqueness regime that is given by the classical Ruelle--Penrose bound by a factor of at least $e$, where the improvements becomes larger as the negative parts of the potential become more prominent (i.e., for attractive interactions at low temperature). | interpretation 0.46 | 0.89 | 0.14 | asserted 1.00 |
| s2 | Our technique is based on the approach of Dyer et al. (Rand. Struct. & Alg. | method 0.99 | 0.41 | 0.02 | asserted 0.95 |
| s3 | '04): we show that for any bounded region and any boundary condition, we can construct a Markov process (in our case spatial birth-death dynamics) that converges rapidly to the finite-volume Gibbs point process while effects of the boundary condition propagate sufficiently slowly. | method 0.97 | 0.87 | 0.04 | asserted 1.00 |
| s4 | As a result, we obtain a spatial mixing property that implies uniqueness of the infinite-volume Gibbs measure. | interpretation 0.54 | 0.92 | 0.29 | asserted 1.00 |
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We prove that for every locally stable and tempered pair potential $\phi$ with bounded range, there exists a unique infinite-volume Gibbs point process on $\mathbb{R}^d$ for every activity $\lambda < (e^{L} \hat{C}_{\phi})^{-1}$, where $L$ is the local stability constant and $\hat{C}_{\phi}:= \mathrm{sup}_{x \in \mathbb{R}^{d}} \int_{\mathbb{R}^{d}} 1 - e^{-|\phi(x, y)|} dy$ is the (weak) temperedness constant. Our result extends the uniqueness regime that is given by the classical Ruelle--Penrose bound by a factor of at least $e$, where the improvements becomes larger as the negative parts of the potential become more prominent (i.e., for attractive interactions at low temperature). Our technique is based on the approach of Dyer et al. (Rand. Struct. & Alg. '04): we show that for any bounded region and any boundary condition, we can construct a Markov process (in our case spatial birth-death dynamics) that converges rapidly to the finite-volume Gibbs point process while effects of the boundary condition propagate sufficiently slowly. As a result, we obtain a spatial mixing property that implies uniqueness of the infinite-volume Gibbs measure.
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We finish our discussion of related uniqueness results by noting that for the special case of purely non-negative potentials, stronger results are known due to Michelen and Perkins [ref] , which improve our uniqueness regime by a factor of at least $$.
Potential-weighted connective constants and uniqueness of Gibbs measures · Comparison to existing bounds for absence of phase transitions · 1.00
Cites as background 1 paper
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As such, it poses an alternative to (much older) analytic notions, such as analyticity of the infinite-volume pressure (see [ref] for a detailed introduction), which are usually proven by studying the convergence of series expansions [ref] or the contraction integral identities [ref] .
Potential-weighted connective constants and uniqueness of Gibbs measures · Introduction · 0.88
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